It took me a while, but here’s what I gather about this (I’m pretty sure it’s correct, but I’m not an expert).
A calibrated forecast means that if you say there is a 20% chance of rain, then it actually rains 20% of the time.
It’s a desired feature, but not the only one (e.g. you could be calibrated by stating: Chick-fil-A is open every day except Monday, but your forecast will always be wrong on Sunday and Monday).
So if
1. you are Bayesian (you state your beliefs)
2. and coherent (the laws of probability apply, so e.g. if P(A) = 0.4, then P(not A) cannot be anything other than 0.6):
and you are predicting something (e.g rain tomorrow), then
if you believe it will rain with probability 0.7 but you are 80% sure of your belief, you won’t say 0.7; you will say something else: 0.8 × 0.7 + 0.2 × something_you_believe = 0.58.
Coherence forces you to collapse your uncertainty into your probability at each forecast.
This theorem shows that, over many forecasts, in your belief system you are certain to be producing a calibrated forecast: your current beliefs assign probability 1 to the proposition that your future forecasts will be calibrated.
But that can’t be, which is the paradox. So Bayesianism is too strong compared to how scientists reason, because scientists always think their model can have an error.
I didn't see a real Bayesian point of view in that article.
A Bayesian does not give you a probability estimate they give you a probability distribution for the probability!
Like in Star Trek Spock is always saying something like "Captain, we have a 15.31% chance of surviving this mission" which is a ridiculous example of precision without accuracy. [1]
If you observe a coin flipped 100 times and it came up heads 65 times it is not a crazy point estimate to say it has a 65% chance of coming up heads but this is just one sample and if you did it another time maybe it comes up 61 or 68 times. You are better saying that the probability distribution of the probability is β(65,35) or maybe β(65.5,35.5) or β(66,36) since that has the "error bars" built in, can be updated if you get more samples, etc.
[1] ... and you know he underestimates survival probabilities the same way Scotty overestimates how long it will take to fix the engines
The \pi_i in the paper is not the estimate of a latent parameter. It is the predictive probability of the event, which is a single number by necessity in a binary challenge. It's the integration of a distribution function which can contains very complex distributions: in my example something_you_believe can be a probability distribution.
So everything in the paper is distribution and when you forecast for a binary event, you give a number which is the expectation of that distribution. This is a probabilistic forecast.
If you were to give a probabilistic forecast for a continuous quantity, then yes you would give in a distribution, as in section 4.2
Spock had not realized that James Kirk emitted a psionic reality distortion field through higher dimensional "luck", if he had he would have been a bit more relaxed.
Something Bayesian. Despite my best effort I just do not get Bayesian probability, it more or less just does not make sense to me. Can you convince me otherwise? What is your best example of something with a probability that can not be analyzed in terms of frequencies or other proportions? And your Bayesian account of it must make sense, I am 90 % certain that P != NP and that is why I would take bets based on those odds does not cut it.
Someone walks out of a magic store holding a coin.
They propose a bet. If they flip it 100 times and the proportion of heads is within [0.4, 0.6], you win $100. If it's not, you pay $100. Do you take that bet?
Explanation: absent the magic store scenario, a `rational' person would take the bet. Your prior belief is that most coins are roughly unbiased. Given that they walked out of a magic store, you now have additional information. Maybe the coin is a trick coin. In that case, your belief that the coin is unbiased should be weaker, even if you don't know which direction the coin is biased in.
This illustrates two things: one, additional information (magic store) can update your beliefs. Two, a strong prior and a weak prior, in this case about the coin's bias, can lead to materially different decisions.
Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?
Let us start with coin flips. You repeatedly flip a coin and the number of heads will come out to be about half the number of trials.
Where does that come from? It is not some intrinsic property of the coin, it comes from varying initial conditions. If you had enough precision when controlling your hand movements, you could in principle force an outcome with high probability.
But assuming you can not or at least do not do that, there is a certain set of initial states, some will lead to heads, some to tails, and each toss will start from a randomly selected initial state. So given my ignorance of the exact initial state, the coin will land heads with a probability equal to the number of initial states leading to heads divided by the number of initial states compatible with my observations of the initial state. [1]
Repeatedly tossing a coin will sample the set of initial states and the result will match the proportion of the number of states. At least as long as I am not wrong about the set of initial states.
The same applies to something like an election. I have imperfect knowledge about the state of the world but there is a set of states compatible with my knowledge about the world and certain subsets of them will lead to certain candidates to win.
[1] Maybe adjusted by some probability distribution over the initial states if they are not equally likely to be picked.
Must ensure that this quote is present whenever Cromwell's Rule is mentioned - "I beseech you, in the bowels of Christ, think it possible that you may be mistaken."
It took me a while, but here’s what I gather about this (I’m pretty sure it’s correct, but I’m not an expert).
A calibrated forecast means that if you say there is a 20% chance of rain, then it actually rains 20% of the time. It’s a desired feature, but not the only one (e.g. you could be calibrated by stating: Chick-fil-A is open every day except Monday, but your forecast will always be wrong on Sunday and Monday).
So if
1. you are Bayesian (you state your beliefs)
2. and coherent (the laws of probability apply, so e.g. if P(A) = 0.4, then P(not A) cannot be anything other than 0.6):
and you are predicting something (e.g rain tomorrow), then if you believe it will rain with probability 0.7 but you are 80% sure of your belief, you won’t say 0.7; you will say something else: 0.8 × 0.7 + 0.2 × something_you_believe = 0.58. Coherence forces you to collapse your uncertainty into your probability at each forecast.
This theorem shows that, over many forecasts, in your belief system you are certain to be producing a calibrated forecast: your current beliefs assign probability 1 to the proposition that your future forecasts will be calibrated.
But that can’t be, which is the paradox. So Bayesianism is too strong compared to how scientists reason, because scientists always think their model can have an error.
I didn't see a real Bayesian point of view in that article.
A Bayesian does not give you a probability estimate they give you a probability distribution for the probability!
Like in Star Trek Spock is always saying something like "Captain, we have a 15.31% chance of surviving this mission" which is a ridiculous example of precision without accuracy. [1]
If you observe a coin flipped 100 times and it came up heads 65 times it is not a crazy point estimate to say it has a 65% chance of coming up heads but this is just one sample and if you did it another time maybe it comes up 61 or 68 times. You are better saying that the probability distribution of the probability is β(65,35) or maybe β(65.5,35.5) or β(66,36) since that has the "error bars" built in, can be updated if you get more samples, etc.
[1] ... and you know he underestimates survival probabilities the same way Scotty overestimates how long it will take to fix the engines
The \pi_i in the paper is not the estimate of a latent parameter. It is the predictive probability of the event, which is a single number by necessity in a binary challenge. It's the integration of a distribution function which can contains very complex distributions: in my example something_you_believe can be a probability distribution.
So everything in the paper is distribution and when you forecast for a binary event, you give a number which is the expectation of that distribution. This is a probabilistic forecast.
If you were to give a probabilistic forecast for a continuous quantity, then yes you would give in a distribution, as in section 4.2
>and you know he underestimates survival probabilities the same way Scotty overestimates how long it will take to fix the engines
NOT SPOCK!!!
Was Spock poorly calibrated?
If he was well calibrated there is no way they would have made it through 79 episodes!
Spock had not realized that James Kirk emitted a psionic reality distortion field through higher dimensional "luck", if he had he would have been a bit more relaxed.
Something Bayesian. Despite my best effort I just do not get Bayesian probability, it more or less just does not make sense to me. Can you convince me otherwise? What is your best example of something with a probability that can not be analyzed in terms of frequencies or other proportions? And your Bayesian account of it must make sense, I am 90 % certain that P != NP and that is why I would take bets based on those odds does not cut it.
Someone walks out of a magic store holding a coin.
They propose a bet. If they flip it 100 times and the proportion of heads is within [0.4, 0.6], you win $100. If it's not, you pay $100. Do you take that bet?
Explanation: absent the magic store scenario, a `rational' person would take the bet. Your prior belief is that most coins are roughly unbiased. Given that they walked out of a magic store, you now have additional information. Maybe the coin is a trick coin. In that case, your belief that the coin is unbiased should be weaker, even if you don't know which direction the coin is biased in.
This illustrates two things: one, additional information (magic store) can update your beliefs. Two, a strong prior and a weak prior, in this case about the coin's bias, can lead to materially different decisions.
Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?
Let us start with coin flips. You repeatedly flip a coin and the number of heads will come out to be about half the number of trials.
Where does that come from? It is not some intrinsic property of the coin, it comes from varying initial conditions. If you had enough precision when controlling your hand movements, you could in principle force an outcome with high probability.
But assuming you can not or at least do not do that, there is a certain set of initial states, some will lead to heads, some to tails, and each toss will start from a randomly selected initial state. So given my ignorance of the exact initial state, the coin will land heads with a probability equal to the number of initial states leading to heads divided by the number of initial states compatible with my observations of the initial state. [1]
Repeatedly tossing a coin will sample the set of initial states and the result will match the proportion of the number of states. At least as long as I am not wrong about the set of initial states.
The same applies to something like an election. I have imperfect knowledge about the state of the world but there is a set of states compatible with my knowledge about the world and certain subsets of them will lead to certain candidates to win.
[1] Maybe adjusted by some probability distribution over the initial states if they are not equally likely to be picked.
What's the probability that the sinking of the USS Maine in 1898 was accidental?
Title a reference to The Well-Tempered Clavier?
(1982)
Must ensure that this quote is present whenever Cromwell's Rule is mentioned - "I beseech you, in the bowels of Christ, think it possible that you may be mistaken."
This is a classic!